文摘

We characterize the \(L^{p}(\sigma )\to L^{q}(\omega )\) boundedness of positive dyadic operators of the form$$T(f\sigma)=\sum\limits_{Q\in\mathcal{D}}\lambda_{Q}{\int}_{Q} f\mathrm{d}\sigma\cdot 1_{Q}, $$and the \(L^{p_{1}}(\sigma _{1})\times L^{p_{2}}(\sigma _{2})\to L^{q}(\omega )\) boundedness of their bilinear analogues, for arbitrary locally finite measures σ,σ₁,σ₂,ω. In the linear case, we unify the existing “Sawyer testing” (for p≤q) and “Wolff potential” (for p>q) characterizations into a new “sequential testing” characterization valid in all cases. We extend these ideas to the bilinear case, obtaining both sequential testing and potential type characterizations for the bilinear operator and all p₁,p₂,q∈(1,∞). Our characterization covers the previously unknown case \(q<\frac {p_{1}p_{2}}{p_{1}+p_{2}}\), where we introduce a new two-measure Wolff potential.KeywordsTwo weight norm inequalityPositive dyadic operatorsDiscrete Wolff potentialTesting conditionsBilinear operatorsFractional integralsT.S.H. and T.P.H. are supported by the European Union through the ERC Starting Grant “Analytic-probabilistic methods for borderline singular integrals”, and are members of the Finnish Centre of Excellence in Analysis and Dynamics Research. K.L. is partially supported by the National Natural Science Foundation of China (11371200), the Research Fund for the Doctoral Program of Higher Education (20120031110023) and the Ph.D. Candidate Research Innovation Fund of Nankai University. This research was conducted during K.L.’s visit to the University of Helsinki.